Transcript Slide 1

Curriculum Focal Points
Mathematics & Science Partnerships
Annual State Coordinators Meeting
June 15, 2006
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Curriculum Focal Points
Charge from the NCTM Board
• To develop a document to guide
curriculum focus at the district and
state levels
• To describe a set of 3-5 curriculum
focal points per grade that serve as a
foundation for a preK-8 curriculum
framework
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Curriculum Focal Points
Three Lenses – the selection process
• Is it mathematically important?
– Preparation for further study
– Use in applications inside and outside of
school
• Does it fit with what is known about
learning mathematics?
• Does it logically connect with the
mathematics in earlier and later grade
levels?
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Curriculum Focal Points
Members of the Writing Team
• Janie Schielack,
Chair
• Doug Clements
• Sharon Lewandowski
• Randy Charles
• Paula Duckett
• Sybilla Beckman
• Emma Trevino
• Rose Zbiek, Chair, NCTM
Essential Understandings
Writing Team
• Francis (Skip) Fennell,
NCTM President Elect, Board
Liaison
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Reviewers
David Bressoud
Bill Bush
Joan Ferrini-Mundy
Anne Collins
Linda Gojak
Jeremy Kilpatrick
Denise Mewborn
Ann Mikesell
Jim Milgram
Barbara Reys
Norm Webb
Mike Shaugnessy
Norma TorresMartinez
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Additional Reviewers - Mathematicians
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Susan Addington
Richard Askey
Tom Banchoff
Hyman Bass
Jerome Dancis
William Dwyer
David Henderson
Rodger Howe
Yorum Sagher
Marjorie Senechal
Maria Terrell
Virginia Warfield
Steve Wilson
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Additional Reviewers – Math Educators
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Michael Battista
Gail Burrill
Cliff Konold
Karen Fuson
Glenda Lappan
Mary Lindquist
Johnny Lott
Gerald Rising
Joe Rosenstein
Susan Jo Russell
Jan Scheer
Grayson Wheatley
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Additional Reviewers – Policy Makers
• William Schmidt
• Cathy Dillender
• Steve Leinwand
• Debbie Nix
• Patsy Wang-Iverson
• Iris Weiss
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Additional Reviewers - Supervisors
• District
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Nancy Acconciamessa
John Carter
Eric Hart
Lisa Kasmer
Jana Palmer
Caroline Piangerelli
Kay Sammons
Dorothy Strong
• State
– Frank Marburger – Pennsylvania
– Donna Watts - Maryland
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Additional Reviewers - Groups
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College Board – via John Dossey
Dallas area – via Dinah Chancellor
EDC – via Paul Goldenberg
Howard County (MD) – via Sharon
Lewandowski
LTCAC Committee – via Gregg McMann
D.C. Teachers – via Paula Duckett
AMTE – via Rose Zbiek
MSDE – via Donna Watts
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Additional Reviewers – NCTM Board
• Jennie Bennett
• Cindy Bryant
• David DeCoste
• Shelley Ferguson
• Bonnie Hagelberger
• Kathy Heid
• Mari Muri
• Richard Seitz
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PreK (4 year olds)
Meaning of Whole Numbers
Attributes
Shapes & Spatial Relationships
K
Representations of Number
Sorting & Classifying
Geometric Shapes & Space
1st
Number Relationships
Meaning of Addition & Subtraction
Shapes
2nd
Place value
Addition & Subtraction (basic number combinations & strategies)
Linear Measurement
3rd
Addition & Subtraction Computation
Multiplication & Division (meaning, basic number combinations &
strategies)
Properties of Polygons
4th
Multiplication Computation
Meaning of Fractions & Decimals
Perimeter & Area
5th
Division Computation
Decimal Computation
Properties of Solids
Representations of Data
6th
Fraction Computation
Meaning of Integers
Meaning of Percent
Volume/Capacity & Surface Area
7th
Ratio, Proportionality & Percent
Integer Computation
Congruence, Similarity &Symmetry
Comparisons of Data Sets
FIRST DRAFT – April 2005
8th
Expressions & Equations
Probability
Geometrical Reasoning & Proof
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Grade Level
Curriculum Focal Points
Grade 3
Number and Operations: Understand
Multiplication & Division Meanings & Develop
Strategies for the Basic Multiplication Facts and
the Related Division Facts
Number & Operations: Understand the Meaning
of Fraction and Fraction Equivalence
Geometry: Describe and Analyze Properties of
Two-Dimensional Shapes
Grade 4
Number & Operations: Develop Immediate
Recall with Multiplication Facts and the Related
Division Facts and Understanding of and Fluency
with a Standard Procedure for Multiplying Whole
Numbers
Number & Operations: Understand the
Meaning of Decimals and Connections Between
Fractions and Decimals
Measurement: Understand and Find the Area of
Two-Dimensional Shapes
Note: title page
FEBRUARY 2006 - DRAFT
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Grade 4 Curriculum Focal Points
Connections to the Focal Points
Number & Operations: Develop Immediate Recall with Multiplication Facts and
the Related Division Facts and Understanding of and Fluency with a Standard
Procedure for Multiplying Whole Numbers
Students use understandings of multiplication to develop immediate recall with basic
multiplication facts and the related division facts. They apply their understanding of models
for multiplication (i.e., equal-sized groups, array and area, number line), place value, and
properties of operations (e.g., distributive property) as they develop, discuss, and use efficient,
and accurate methods to multiply multi-digit whole numbers. They select and accurately
apply appropriate methods to estimate products and mentally calculate products depending
upon the context and the numbers involved. They develop fluency with standard procedures
for multiplication of whole numbers through three-digit by two-digit numbers.
Algebra Readiness: Students continue
identifying, describing and extending numeric
patterns involving all operations, and non-numeric
growing or repeating patterns. Properties of
multiplication are revisited.
Number & Operations: Understand the Meaning of Decimals and Connections
Between Fractions and Decimals
Students understand that decimal notation is the use of the base-ten numeration system
to represent a number, including numbers between 0 and 1, between 1 and 2, and so forth.
Students extend their understandings of fractions to reading and writing decimals greater than
or less than one, identifying equivalent decimals, comparing and ordering decimals, and
estimating decimal/fractional amounts. They connect equivalent fractions and decimals by
comparing models and symbols and locating them together on the number line.
Measurement: Understand and Find the Area of Two-Dimensional Shapes
Students recognize area as an attribute of the region enclosed by a two-dimensional
shape. They understand that area can be quantified by finding the total number of same-size
units of area it takes to cover the shape without gaps or overlap. They understand that a 1unit by 1-unit square is the standard unit for measuring area. They select appropriate units,
strategies (e.g., decomposing shapes), and tools for estimating and measuring area. Students
connect area measure to the area model used to represent multiplication, and they use this
connection to justify and find relationships among the formulas for the areas of different
polygons and solve problems involving area. They measure necessary attributes of shapes in
order to use an area formula.
FEBRUARY 2006 - DRAFT
Geometry: Properties of two-dimensional shapes
are revisited from Grade 3 as students find areas
of polygons at Grade 4. Through the design and
analysis of simple tilings and tessellations,
students deepen their understanding of twodimensional space including measuring and
classifying angles. Students classify and measure
angles.
Data Analysis: Frequency tables, bar graphs,
picture graphs, and line plots are continued from
Grade 3 as problem contexts and tools for solving
problems. Students apply their understanding of
place value to develop stem and leaf plots and
use them to solve problems.
Number and Operations: Students develop
understanding of strategies for multi-digit division
using models based on division as the inverse of
multiplication, division as partitioning, and division
as successive subtraction.
Continued Development in Number and
Operations: Students extend their understanding
of place value and ways of representing numbers
with numbers to 100,000 in various contexts.
Fraction equivalence is revisited in the work with
decimals. Students’ work with multiplication and
division facts is applied to techniques for
simplifying fractions.
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Connections to the Focal Points
FINAL
Grade 4 Curriculum Focal Points
Number and Operations and Algebra: Developing Quick Recall of Multiplication Facts and
Related Division Facts and Fluency with Whole Number Multiplication
Students use understandings of multiplication to develop quick recall of the basic multiplication
facts and related division facts. They apply their understanding of models for multiplication
(i.e., equal-sized groups, arrays, area models, equal intervals on the number line), place value,
and properties of operations, in particular the distributive property, as they develop, discuss, and
use efficient, accurate, and generalizable methods to multiply multi-digit whole numbers. They
select and accurately apply appropriate methods to estimate products and mentally calculate
products depending upon the context and the numbers involved. They develop fluency with
efficient procedures, including the standard algorithm, for multiplying whole numbers;
understand why the procedures work based on place value and properties of operations; and use
them to solve problems.
Number and Operations: Developing Understanding of Decimals, including Connections
Between Fractions and Decimals
Students understand that decimal notation is the use of the base-ten numeration system to
represent a number, including numbers between 0 and 1, between 1 and 2, and so forth.
Students extend their understanding of fractions to reading and writing decimals greater than or
less than one, identifying equivalent decimals, comparing and ordering decimals, and
estimating decimal/fractional amounts in problem-solving situations. They connect equivalent
fractions and decimals by comparing models and symbols and locating them together on the
number line.
Measurement: Developing Understanding of Area and Determining the Areas of TwoDimensional Shapes
Students recognize area as an attribute of two-dimensional regions. They understand that area
can be quantified by finding the total number of same-size units of area required to cover the
shape without gaps or overlap. They understand that a 1-unit by 1-unit square is the standard
unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes),
and tools for solving problems that involve estimating and measuring area. Students connect
area measure to the area model used to represent multiplication, and they use this connection to
justify the formula for finding the area of a rectangle.
Algebra: Students continue identifying,
describing and extending numeric patterns
involving all operations and non-numeric
growing or repeating patterns in order to
develop understanding of the use of a rule to
describe a sequence of numbers or objects.
Geometry: Students extend their
understanding of properties of two-dimensional
shapes as they find areas of polygons. They
extend their work with symmetry and
congruence in Grade 3 to include
transformations, including those that produce
line and rotational symmetry. Through the use
of transformations to design and analyze simple
tilings and tessellations, students deepen their
understanding of two-dimensional space.
Measurement: As part of understanding twodimensional shapes, students measure and
classify angles.
Data Analysis: Frequency tables, bar graphs,
picture graphs, and line plots are continued
from Grade 3 as problem contexts and tools for
solving problems. Students apply their
understanding of place value to develop and use
stem-and-leaf plots.
Number and Operations: Based on Grade 3
work, students extend their understanding of
place value and ways of representing numbers
to 100,000 in various contexts. They use
estimation in determining the relative size of an
amount or distance. Students develop
understandings of strategies for multi-digit
division using models based on division as the
inverse of multiplication, division as
partitioning, and division as successive
subtraction. Fraction equivalence is extended
in the work with decimals. Students’ work with
multiplication and division facts in Grade 3 is
applied to techniques for simplifying fractions.
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Consider – from the Preface
• Organizing curriculum around focal points,
with a clear emphasis on the processes…,
and, particularly problem solving…
• Immediate uses…stimulate discussion
• Long term opportunity…starting point for
continued dialogue and discussion…next
generation of curriculum standards, tests,
and textbooks.
• Use them, discuss them, argue about them,
refine them.
Page ii
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Consider – Why Identify Focal Points?
• What mathematics should be the
focus of instruction and student
learning at the PreK-8 level?
• A Quest for Coherence provides one
possible response to the question of
how to organize curriculum…
• …wide variety of mathematics
curriculum standards, little consensus
(see Reys, et al).
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Number of 4th grade learning expectations
per state by content strand
Number
&
Operations
Geometry
Measurement
Algebra
Data Analysis,
Probability & Statistics
Total Number of LEs
California
16
11
4
7
5
43
Texas
15
7
3
4
3
32
New York
27
8
10
5
6
56
Florida
31
11
17
10
20
89
Ohio
15
8
6
6
13
48
Michigan
37
5
11
0
3
56
New Jersey
21
10
8
6
11
56
North Carolina
14
3
2
3
4
26
Georgia
23
10
5
3
4
45
Virginia
17
8
11
2
3
41
Reys, et al, 2006
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Consider – What are CFPs?
• …represent a set of important
mathematical topics for each level…
• …organizing structures, key topics,
foundations for further mathematics
learning…
• …situated within applications of the
processes one view…, which is
different from a set of grade-level
mastery objectives..
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Consider – How Should CFPs be used?
• …directed to those who are responsible for
development of mathematics curriculum…
• This document is presented as an example
from which the next generation…might be
built.
• …identifying for curriculum developers
grade level targets…
• This document does not include instructional
approaches…
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Consider – How related…to PSSM?
• …the curriculum focal points are
targeted mostly toward content. The
mathematical processes described
within PSSM are expected to be
addressed in the implemented
curriculum through…to discuss and
validate their mathematical
thinking…and apply the mathematics
they are learning to solve problems…
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Consider – How related…to PSSM? (cont.)
• The integrated nature of the CFPs is
highlighted…many relate to more than one
content strand.
• Connections to the focal points…serve two
purposes:
– Recognize the need for introductory and
continuing experiences related to focal points
that may be identified at other grade levels.
– Identify ways in which a grade-level’s focal
points can be used to support learning in relation
to strands that are not focal points at that grade
level.
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Why is this important?
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Why should NCTM do this?
• Curricular Coherence
• Need (see next two slides)
• Opportunity
– National Mathematics Panel
– ACI
• Math Now – next year
– PACE
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Recommendations:
Specification of Learning Expectations
• Identify major goals or focal points at each
grade level. At each grade we
recommend a general statement of major
goals for the grade be stated. These goals
may emphasize a few strands of
mathematics or a few topics within strands.
• …This may also help reduce superficial
treatment of many mathematical topics, a
common criticism of the U.S. mathematics
curriculum.
Reys, et al, in press.
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Recommendations:
Collaborate to Promote Consensus
• …states will need to work together to create
some level of consensus about important
curriculum goals at each grade.
• …if states build their curriculum standards
from a “core curriculum” offered by national
groups such as the NCTM, College Board,
and/or Achieve.”
Reys et al, in press26
And…
• Provide guidance and resources for establishing
and enacting mathematics curriculum that is
coherent, focused, well-articulated and consistent
with PSSM (curriculum).
• Engage in political and public advocacy to focus
decision makers on improving learning and
teaching mathematics (advocacy)
• Advance professional development by creating a
coherent framework of audience-specific products
and services (professional development)
NCTM’s - STRATEGIC PLANNING OBJECTIVES
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The Importance
of
Big Picture Thinking
Basic Mathematical Skills (NCSM, 1977)
An Agenda for Action (NCTM, 1980)
Curriculum & Evaluation Standards (NCTM, 1989)
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Questions?
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Motion to:
• Approve the PreK-8 Curriculum Focal
Points as a position of the National
Council of Teachers of Mathematics.
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Questions
Next Steps
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Francis (Skip) Fennell
[email protected] OR [email protected]
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